A Crossing Lemma for Jordan curves

János Pach, Natan Rubin, G. Tardos

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(log⁡log⁡(|T|/n))1/504). In particular, if |T|/n→∞ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1−o(1))n2.

Original languageEnglish
Pages (from-to)908-940
Number of pages33
JournalAdvances in Mathematics
Volume331
DOIs
Publication statusPublished - Jun 20 2018

Fingerprint

Jordan Curve
Lemma
Intersection
Intersection of sets
Curve
Pairwise
Corollary
Closed

Keywords

  • Arrangements of curves
  • Combinatorial geometry
  • Contact graphs
  • Crossing Lemma
  • Extremal problems
  • Separators

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A Crossing Lemma for Jordan curves. / Pach, János; Rubin, Natan; Tardos, G.

In: Advances in Mathematics, Vol. 331, 20.06.2018, p. 908-940.

Research output: Contribution to journalArticle

Pach, János ; Rubin, Natan ; Tardos, G. / A Crossing Lemma for Jordan curves. In: Advances in Mathematics. 2018 ; Vol. 331. pp. 908-940.
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